Nested Apparent Horizons and Quantized Separation Demonstrate Hawking Backreaction Effects

The intense radiation emitted by black holes, known as Hawking radiation, fundamentally alters the surrounding spacetime when it becomes strong enough to influence the black hole itself. Steven J. Silverman from UCLA Samueli School of Engineering and colleagues demonstrate that this powerful outgoing energy can create multiple, nested boundaries around a black hole, effectively building layers of event horizons. This research reveals that these horizons not only form but can also merge, a process reminiscent of energy extraction predicted by the Penrose process, yet driven by the back reaction of the radiation itself. Importantly, the team proposes that the distance separating these nested horizons may be discrete, suggesting a geometric pathway towards understanding how fundamental quantities in physics might be quantized, a concept with profound implications for our understanding of spacetime and gravity.

Dynamical Spacetime and Nested Apparent Horizons

In general relativity, standard black hole evaporation models assume emitted energy negligibly affects spacetime geometry. However, when emission rates are extreme, such as near the end of evaporation or close to a rotating black hole’s ergosphere, the outgoing energy density becomes comparable to the curvature scale, demanding a fully dynamical treatment. In such scenarios, distinguishing between a global event horizon and local apparent or trapping horizons becomes essential. Researchers explore how strong outgoing radiation can generate a second apparent horizon exterior to the original, creating a transient “nested-horizon” configuration.

The Vaidya metric provides a convenient framework for modelling dynamical, spherically symmetric spacetimes with outgoing energy flux. This metric describes spacetime in terms of retarded time and the Bondi mass, representing the total energy observed at infinity. The associated stress tensor describes outgoing null radiation. Local trapping surfaces, defining apparent horizons, are identified by the vanishing of the outgoing null expansion. This condition yields the apparent-horizon radius, determined by the Misner, Sharp mass inside a given radius.

When the flux is weak, the Misner, Sharp mass increases monotonically with radius, resulting in a single apparent horizon. However, a compact, energetic shell of outgoing radiation can deform the Misner, Sharp mass, allowing the apparent-horizon condition to admit multiple roots, corresponding to nested apparent horizons. Scientists consider a continuous mass profile representing a black hole surrounded by an outgoing null shell of energy to illustrate this behaviour. For small energy shells, a single root persists. As the shell energy increases beyond a critical value, the relationship yields two or three intersections, representing an inner and an outer trapped surface.

The outer root roughly follows a predictable relationship, consistent with a simplified step-function model. As the shell propagates outward, the outer apparent horizon expands and eventually vanishes, leaving only the inner horizon. In a fully time-dependent picture, these surfaces merge and annihilate, consistent with numerical studies of dynamical horizons. If horizon area and surface gravity are adiabatic invariants, the separation between successive apparent horizons may be subject to a discrete spectrum, analogous to quantized orbital radii in atomic physics. Following Bekenstein’s adiabatic-invariant argument, researchers postulate a Bohr, Sommerfeld-type condition relating the surface gravity, horizon area, and quantum numbers.

This yields a discrete spectrum of allowed areas, with the Planck length as a fundamental scale. If a radiating black hole temporarily supports two apparent horizons, their areas could satisfy a specific relationship, imposing a quantum restriction on the possible radial separation. Thus, the nested horizons behave as discrete “gravitational shells,” with quantized spacing analogous to electronic orbitals. Between the inner and outer horizons, null geodesics experience an effective potential associated with the local Misner, Sharp mass function. A semiclassical WKB quantization condition would then determine the discrete, metastable configurations where outgoing Hawking flux and gravitational confinement balance.

The inner and outer horizons form a self-consistent “quantum cavity” for vacuum modes, whose resonant states could manifest as discrete energy or area levels. This interpretation connects with several strands of quantum-gravity research, including loop quantum gravity and string-inspired models, which also predict quantized horizon microstates. If horizon separations are indeed quantized, the smallest possible separation would be of order the Planck length, establishing a minimal “horizon spacing” and suggesting that the evaporation endpoint corresponds to a finite, Planck-scale remnant. The presence of transient nested apparent horizons does not imply two event horizons, as the event horizon is a global construct.

Nevertheless, such transient trapped regions could influence the local thermodynamics and entropy bookkeeping of highly radiative black holes, and may provide a bridge toward models of regular or bouncing geometries. In the extreme regime where the Hawking flux itself sources significant curvature, a quantum-gravitational treatment will be necessary. The model captures the essential qualitative mechanism by which strong backreaction can momentarily cloak an evaporating black hole within a second, dynamically generated apparent horizon.

Nested Horizons Formed by Hawking Radiation

This work demonstrates that intense Hawking radiation emitted from both rotating and non-rotating black holes can create transient, nested apparent horizons exterior to the original event horizon. Researchers employed a spherically symmetric semiclassical model to show that strong outgoing energy flux generates these additional horizons, reminiscent of the Penrose process but driven by back reaction. The study reveals that under certain conditions, these horizons bifurcate and merge, creating complex configurations of trapped surfaces. For a black hole of a specific mass and initial radius, with a small energy perturbation, the team observed that for low shell energies, a single horizon exists.

However, as the shell energy increases beyond a critical value, two additional roots appear, signifying the formation of a transient outer horizon due to the intense outgoing flux. Further analysis proposes a discrete quantization rule for the separation between these nested horizons, analogous to the Bohr-Sommerfeld condition. This yields a discrete spectrum of allowed areas, with a separation defined by a specific equation involving fundamental constants. Consequently, the radial separation between horizons is quantified, suggesting that these nested horizons behave as discrete “gravitational shells” with quantized spacing.

The team also explored a semiclassical WKB quantization condition, revealing that the inner and outer horizons form a self-consistent “quantum cavity” for vacuum modes, potentially leading to discrete energy or area levels. This research connects to several quantum-gravity proposals, including loop quantum gravity and string-inspired models, which also predict quantized horizon microstates. The study suggests that the smallest possible horizon separation is on the order of the Planck length, establishing a minimal horizon spacing and implying that black hole evaporation may terminate in a finite, Planck-scale remnant. While direct detection remains beyond current capabilities, the nested-horizon framework offers a potential bridge between classical backreaction dynamics and quantum discreteness, potentially identifying the geometric origin of Planck-scale quantization and the microscopic basis of black hole entropy.

Hawking Radiation Creates Transient Horizons

This research demonstrates that intense Hawking radiation emitted by black holes can create temporary, nested apparent horizons outside the original event horizon, a phenomenon arising from the back reaction of the emitted energy. Using a simplified model, scientists show that strong outgoing energy flux can generate these multiple horizons, reminiscent of the Penrose process but driven by the radiation itself. The study derives conditions under which these horizons bifurcate and merge, suggesting a potential connection to discrete quantization rules analogous to those found in atomic physics. The findings offer a potential bridge between classical descriptions of black hole dynamics and the quantum realm, potentially illuminating the geometric origin of Planck-scale quantization and the microscopic basis of black hole entropy. While direct detection of the predicted quantization remains beyond current experimental capabilities, the researchers suggest analogous features might be observable in numerical simulations of quantum collapse or within the spectra of quasinormal modes.